Cho P=căn x/căn x-1+3/căn x+1-6 căn x-4/x-1 . Tìm tất cả các giá trị của sao cho P<1/2

* Hướng dẫn giải

$\begin{array}{l}P=\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{3}{\sqrt{x}+1}-\frac{6\sqrt{x}-4}{x-1}\left(\begin{array}{l}x\ge 0\\ x\ne 1\end{array}\right)\\ =\frac{\sqrt{x}\left(\sqrt{x}+1\right)+3\left(\sqrt{x}-1\right)-6\sqrt{x}+4}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{x+\sqrt{x}+3\sqrt{x}-3-6\sqrt{x}+4}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\\ =\frac{x-2\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{{\left(\sqrt{x}-1\right)}^2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}-1}{\sqrt{x}+1}\end{array}$

$\begin{array}{l}P<\frac{1}{2}\iff \frac{\sqrt{x}-1}{\sqrt{x}+1}<\frac{1}{2}\iff \frac{\sqrt{x}-1}{\sqrt{x}+1}-\frac{1}{2}<0\iff \frac{2\sqrt{x}-2-\sqrt{x}-1}{2\left(\sqrt{x}+1\right)}<0\\ \Rightarrow \sqrt{x}-3<0\Rightarrow \sqrt{x}<3\Rightarrow x<9\Rightarrow \left\{\begin{array}{l}0\le x<9\\ x\ne 1\end{array}\right.\end{array}$